Index of a subgroup


In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G : H| or or.
Formally, the index of H in G is defined as the number of cosets of H in G. For example, let Z be the group of integers under addition, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z, so the index of 2Z in Z is two. To generalize,
for any positive integer n.
If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G / N, since this is defined in terms of a group structure on the set of cosets of N in G.
If G is infinite, the index of a subgroup H will in general be a non-zero cardinal number. It may be finite - that is, a positive integer - as the example above shows.
If G and H are finite groups, then the index of H in G is equal to the quotient of the orders of the two groups:
This is Lagrange's theorem, and in this case the quotient is necessarily a positive integer.

Properties

If H has an infinite number of cosets in G, then the index of H in G is said to be infinite. In this case, the index |G : H| is actually a cardinal number. For example, the index of H in G may be countable or uncountable, depending on whether H has a countable number of cosets in G. Note that the index of H is at most the order of G, which is realized for the trivial subgroup, or in fact any subgroup H of infinite cardinality less than that of G.

Finite index

An infinite group G may have subgroups H of finite index. Such a subgroup always contains a normal subgroup N, also of finite index. In fact, if H has index n, then the index of N can be taken as some factor of n!; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left cosets of H.
A special case, n = 2, gives the general result that a subgroup of index 2 is a normal subgroup, because the normal subgroup must have index 2 and therefore be identical to the original subgroup. More generally, a subgroup of index p where p is the smallest prime factor of the order of G is necessarily normal, as the index of N divides p! and thus must equal p, having no other prime factors.
An alternative proof of the result that subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in.

Examples

The above considerations are true for finite groups as well. For instance, the group O of chiral octahedral symmetry has 24 elements. It has a dihedral D4 subgroup of order 8, and thus of index 3 in O, which we shall call H. This dihedral group has a 4-member D2 subgroup, which we may call A. Multiplying on the right any element of a right coset of H by an element of A gives a member of the same coset of H. A is normal in O. There are six cosets of A, corresponding to the six elements of the symmetric group S3. All elements from any particular coset of A perform the same permutation of the cosets of H.
On the other hand, the group Th of pyritohedral symmetry also has 24 members and a subgroup of index 3, but in this case the whole subgroup is a normal subgroup. All members of a particular coset carry out the same permutation of these cosets, but in this case they represent only the 3-element alternating group in the 6-member S3 symmetric group.

Normal subgroups of prime power index

Normal subgroups of prime power index are kernels of surjective maps to p-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
As these are weaker conditions on the groups K, one obtains the containments
These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

Geometric structure

An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the complement of their symmetric difference yields a third. This is a simple corollary of the above discussion.
However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index p form a projective space, namely the projective space
In detail, the space of homomorphisms from G to the group of order p, is a vector space over the finite field A non-trivial such map has as kernel a normal subgroup of index p, and multiplying the map by an element of does not change the kernel; thus one obtains a map from
to normal index p subgroups. Conversely, a normal subgroup of index p determines a non-trivial map to up to a choice of "which coset maps to which shows that this map is a bijection.
As a consequence, the number of normal subgroups of index p is
for some k; corresponds to no normal subgroups of index p. Further, given two distinct normal subgroups of index p, one obtains a projective line consisting of such subgroups.
For the symmetric difference of two distinct index 2 subgroups gives the third point on the projective line containing these subgroups, and a group must contain index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.